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More on Regular Subgroups of the Affine Group 3 MORE ON REGULAR SUBGROUPS OF THE AFFINE GROUP M.A. PELLEGRINI AND M.C. TAMBURINI BELLANI Abstract. This paper is a new contribution to the study of regular subgroups of the affine group AGLn(F), for any field F. In particular we associate to any partition λ =6 (1n+1) of n + 1 abelian regular subgroups in such a way that different partitions define non-conjugate subgroups. Moreover, we classify the regular subgroups of certain natural types for n ≤ 4. Our classification is equivalent to the classification of split local algebras of dimension n + 1 over F. Our methods, based on classical results of linear algebra, are computer free. 1. Introduction Let F be any field. We identify the affine group AGLn(F) with the subgroup of T GLn+1(F) consisting of the matrices having (1, 0,..., 0) as first column. With this n notation, AGLn(F) acts on the right on the set = (1, v): v F of affine points. A { ∈ } Clearly, there exists an epimorphism π : AGLn(F) GLn(F) induced by the action n → of AGLn(F) on F . A subgroup (or a subset) R of AGLn(F) is called regular if it acts regularly on , namely if, for every v Fn, there exists a unique element in R A ∈ having (1, v) as first row. Thus R is regular precisely when AGLn(F)= GLn(F) R, with GLn(F) R = In , where GLn(F) denotes the stabilizer of (1, 0,..., 0). ∩ { +1} A subgroup H of AGLn(F) is indecomposable if there exists no decompositionc of Fn asc a direct sum of non-trivial π(cH)-invariant subspaces. Clearly, to investigate the structure of regular subgroups, the indecomposable ones are the most relevant, since the other ones are direct products of regular subgroups in smaller dimensions. So, one has to expect very many regular subgroups when n is big. Actually, in Section 6 we show how to construct at least one abelian regular subgroup, called standard, for each partition λ = (1n+1) of n + 1, in such a way that different partitions produce non-conjugate6 subgroups. Several of them are indecomposable. The structure and the number of conjugacy classes of regular subgroups depend F F arXiv:1601.01679v2 [math.GR] 14 Jan 2016 on . For instance, if has characteristic p> 0, every regular subgroup is unipotent n+1 [14, Theorem 3.2], i.e., all its elements satisfy (t In+1) = 0. A unipotent group is conjugate to a subgroup of the group of upper− unitriangular matrices (see [10]): in particular it has a non-trivial center. By contrast to the case p > 0, AGL2(R) R contains 2| | conjugacy classes of regular subgroups with trivial center, hence not unipotent (see Example 2.5). So, clearly, a classification in full generality is not realistic. Since the center Z(R) of a regular subgroup R is unipotent (see Theorem 2.4(a)) if Z(R) is non-trivial one may assume, up to conjugation, that R is contained in the centralizer of a unipotent Jordan form (see Theorem 4.4). But even this condition is weak. 2010 Mathematics Subject Classification. 15A21, 20B35, 16L99. Key words and phrases. Regular subgroup; local algebra; Jordan canonical form. 1 2 M.A.PELLEGRINIANDM.C.TAMBURINIBELLANI Before introducing a stronger hypothesis, which allows to treat significant cases, we need some notation. We write every element r of R as 1 v 1 v 1 v (1) r = = = = µR(v), 0 π(r) 0 τR(v) 0 In + δR(v) n n n where µR : F AGLn(F), τR : F GLn(F) and δR := τR id : F Matn(F). → → − → The hypothesis we introduce is that δR is linear. First of all, if R is abelian, then δR is linear (see [1]). Moreover, if δR is linear, then R is unipotent by Theorem 2.4(b), but not necessarily abelian. One further motivation for this hypothesis is that δR is linear if and only if = FIn + R is a split local subalgebra of L +1 Matn+1(F). Moreover, two regular subgroups R1 and R2, with δRi linear, are conjugate in AGLn(F) if and only if the corresponding algebras 1 and 2 are isomorphic (see Section 3). In particular, there is a bijection betweL en conjugacyL classes of abelian regular subgroups of AGLn(F) and isomorphism classes of abelian split local algebras of dimension n + 1 over F. This fact was first observed in [1]. It was studied also in connection with other algebraic structures in [3, 4, 5, 6] and in [2], where the classification of nilpotent associative algebras given in [7] is relevant. In Section 7 we classify, up to conjugation, certain types of regular subgroups U of AGLn(F), for n 4, and the corresponding algebras. More precisely, for n = 1 the only regular≤ subgroup is the translation subgroup , which is standard. T For n = 2, 3, we assume that δU is linear. If n = 2 all subgroups U, over any F, are standard (Table 1). For n = 3 the abelian regular subgroups are described in Table 2. The non abelian ones are determined in Lemmas 7.2 and 7.3: there are F conjugacy classes when char F = 2, F + 1 otherwise (see also [7]). If n = 4 we| | assume that U is abelian. The conjugacy| | classes, when F has no quadratic extensions, are shown in Tables 3 and 4. In particular, by the reasons mentioned above, we obtain an independent clas- sification of the split local algebras of dimension n 4 over any field. We obtain also the classification of the commutative split local≤ algebras of dimension 5 over fields with no quadratic extensions. Actually, regular subgroups arise from matrix representations of these algebras. In the abelian case and using the hypothesis that F is algebraically closed, the same classification, for n 6, was obtained by Poonen [12], via commutative algebra. Namely he presents≤ the algebras as quotients of the polynomial ring F[t1,...,tr], r 1. The two approaches are equivalent but, comparing our results with those of≥ Poonen, we detected an inac- curacy1. Namely for n = 5 and char F = 2, the two algebras defined, respectively, by F[x,y,z]/ x2,y2,xz,yz,xy + z2 and F[x,y,z]/ xy,xz,yz,x2 + y2, x2 + z2 are isomorphic (seeh Remark 7.6). i h i Our methods, based on linear algebra, are computer independent. 2. Some basic properties and examples of regular subgroups Lemma 2.1. A regular submonoid R of AGLn(F) is a subgroup. n 1 Proof. For any v F , the first row of µR vτR(v)− µR(v) is the same as the ∈ − 1 first row of In+1. From the regularity of R it follows µR vτR(v)− µR(v)= In+1, 1 1 − whence µR(v)− = µR vτR(v)− R. Thus R is a subgroup. − ∈ 1See [13] for a revised version of [12]. MORE ON REGULAR SUBGROUPS OF THE AFFINE GROUP 3 n If δR HomZ (F , Matn(F)), i.e. δR is additive, direct calculation gives that a ∈ regular subset R of AGLn(F) containing In+1 is a submonoid, hence a subgroup, if and only if: n (2) δR(vδR(w)) = δR(v)δR(w), for all v, w F . ∈ Also, given v, w Fn, ∈ (3) µR(v)µR(w)= µR(w)µR(v) if and only if vδR(w)= wδR(v). For the next two theorems we need a basic result, that we recall below for the reader’s convenience. A proof can be found in any text of linear algebra. Lemma 2.2. Let g GLm(F) have characteristic polynomial χg(t) = f (t)f (t). ∈ 1 2 If (f1(t),f2(t)) = 1, then g is conjugate to h = diag(h1,h2), with χhi (t) = fi(t), i =1, 2. Also, CMatm(F)(h) consists of matrices of the same block-diagonal form. Lemma 2.3. Let z be an element of a regular subgroup R of AGLn(F). If z is not unipotent then, up to conjugation of R under AGLn(F), we may suppose that 1 w1 0 (4) z = 0 A 0 , 1 0 0 A2 where A1 is unipotent and A2 does not have the eigenvalue 1. 1 w Proof. Let z = , where z = I + δ (w). Since z has the eigenvalue 1 [14, 0 z 0 n R 0 0 Lemma 2.2], the characteristic polynomial χz0 (t) of z0 factorizes in F[t] as m1 χz (t) = (t 1) g(t), g(1) =0, m 1, deg g(t)= m 1. 0 − 6 1 ≥ 2 ≥ By Lemma 2.2, up to conjugation in GLn(F), we may set z0 = diag(A1, A2) with: m1 χA (t) = (t 1) , χA (t)= g(t). 1 − c 2 Write w = (w , w ) with w Fm1 and w Fm2 . Since A does not have the 1 2 1 ∈ 2 ∈ 2 eigenvalue 1, the matrix A2 Im2 is invertible. Conjugating by the translation 1 − 1 0 w (A Im )− 1 w 0 2 2 − 2 1 0 Im 0 we may assume w = 0, i.e. z = 0 A 0 . 1 2 1 0 0 Im2 0 0 A2 n We observe that if δR HomF (F , Matn(F)), i.e. δR is linear, then there is a ∈ natural embedding of R into a regular subgroup R of AGLn(F) for any extension F of F. Namely: b b 1v ˆ n b R = :v ˆ F F F AGLn(F), 0 In + δ b(ˆv) ∈ ⊗ ≤ R b b b where δ b λvˆ i = λˆiδR(vi), λˆi F.
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